Topological Methods in Nonlinear Analysis

Periodic solutions for the non-local operator $(-\Delta+m^{2})^{s}-m^{2s}$ with $m\geq 0$

Vincenzo Ambrosio

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


By using variational methods, we investigate the existence of $T$-periodic solutions to \begin{equation*} \begin{cases} [(-\Delta_{x}+m^{2})^{s}-m^{2s}]u=f(x,u) &\mbox{in } (0,T)^{N}, \\ u(x+Te_{i})=u(x) &\mbox{for all } x \in \mathbb{R}^N, \ i=1, \dots, N, \end{cases} \end{equation*} where $s\in (0,1)$, $N> 2s$, $T> 0$, $m\geq 0$ and $f$ is a continuous function, $T$-periodic in the first variable, verifying the Ambrosetti-Rabinowitz condition, with a polynomial growth at rate $p\in (1, ({N+2s})/({N-2s}))$.

Article information

Topol. Methods Nonlinear Anal., Volume 49, Number 1 (2017), 75-103.

First available in Project Euclid: 11 April 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Ambrosio, Vincenzo. Periodic solutions for the non-local operator $(-\Delta+m^{2})^{s}-m^{2s}$ with $m\geq 0$. Topol. Methods Nonlinear Anal. 49 (2017), no. 1, 75--103. doi:10.12775/TMNA.2016.063.

Export citation


  • A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381.
  • V. Ambrosio, Existence of heteroclinic solutions for a pseudo-relativistic Allen–Cahn type equation, Adv. Nonlinear Stud. 15 (2015), 395–414.
  • ––––, Periodic solutions for a pseudo-relativistic Schrödinger equation, Nonlinear Anal. 120 (2015), 262–284.
  • D. Applebaum, Lévy processes and stochastic calculus, Cambridge Stud. Adv. Math. 93 (2004).
  • A. Bènyi and T. Oh, The Sobolev inequality on the torus revisited, Publ. Math. Debrecen 83 (2013), no. 3, 359–374.
  • P. Biler, G. Karch and W.A. Woyczynski, Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws, Ann. Inst. H. Poincaré Anal. Non Linéaire 18 (2001), 613–637.
  • X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians \romI: regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), 23–53.
  • ––––, Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions, Trans. Amer. Math. Soc. 367 (2015), 911–941.
  • X. Cabré and J. Solà-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math. 58 (2005), 1678–1732.
  • X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math. 224 (2010), 2052–2093.
  • L.A. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), 1245–1260.
  • L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. 2 171 (2010), no. 3, 1903–1930.
  • A. Capella, J. Dávila, L. Dupaigne and Y. Sire, Regularity of radial extremals solutions for some non-local semilinear equation, Comm. Partial Differential Equations 36 (2011), 1353–1384.
  • R. Carmona, W.C. Masters and B. Simon, Relativistic Schrödinger operators; Asymptotic behaviour of the eigenfunctions, J. Funct. Anal. 91 (1990), 117–142.
  • R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman and Hall/CRC Financ. Math. Ser., Chapman and Hall/CRC, Boca Raton, FL (2004).
  • G. Duvaut and J.L. Lions, Inequalities in Mechanics and Physics, Grundlehren Math. Wiss., vol. 219, Springer–Verlag, Berlin, 1976. Transl. from French by C.W. John.
  • A. Erdélyi, W. Magnus, F. Oberhettinger and F. Tricomi, Higher Trascendental Functions, vol. 1,2 McGraw-Hill, New York (1953).
  • M.M. Fall and V. Felli, Unique continuation properties for relativistic Schrödinger operators with a singular potential, DCDS A. (to appear).
  • J. Fröhlich, B.L.G. Jonsson and E. Lenzmann, Boson stars as solitary waves, Comm. Math. Phys. 274 (1), 1–30.
  • E.H. Lieb and M. Loss, Analysis, 2nd Edition. Vol. 14 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI. 33.
  • E.H. Lieb and H.T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Comm. Math. Phys. 112 (1987), 147–174.
  • ––––, The stability and instability of relativistic matter, Comm. Math. Phys. 118 (2), (1988) 177–213.
  • J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences, Springer–Verlag, New York 74 (1989).
  • P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics 65 (1986).
  • M. Ryznar, Estimate of Green function for relativistic $\alpha$-stable processes, Potential Analysis 17 (2002), 1–23.
  • L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math. 60 (2006), 67–112.
  • Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, J. Funct. Anal. 256 (2009), 1842–1864.
  • J.J. Stoker, Water Waves: The Mathematical Theory with Applications, Pure Appl. Math., vol. IV, Interscience Publishers, Inc., New York, (1957).
  • M. Struwe, Variational methods: Application to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer–Verlag, Berlin, (1990).
  • J.F. Toland, The Peierls–Nabarro and Benjamin–Ono equations, J. Funct. Anal. 145 (1997), 136–150.
  • M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications 24 (1996).
  • A. Zygmund, Trigonometric Series, Vol. 1, 2, Cambridge University Press, Cambridge (2002).